. 7 - 151 



Which shows that the velocity of the centre of the buoble varies inversely as the cube of the 

 radius of the DubBle, and therefore increases greatly as the bubble contracts. The motion Is In 

 fact the same as that of a particle, whose mass Is variable and proportional to the volume of the 

 bubble. 



If we write 



0^ = b^(t)/b (0) (33) 



and regjrd-/? as n function ;f a, we obtain from (32) and (17) 



0^1 . , L___ (3^, 



da aV{ a-5 - a-^ ♦ (r - I) P-' (a-5 - 1) } 



wtnre 



Hence 



a{0) /3 fy- 1) 



(35) 



^1 J a^/{a^ - a^ ♦ (y - i) p^ Q (a^ - i)> 



and /3, is therefore an inverse-periodic function of a. 



When 7 = 4/3, the integral may be evaluated In exactly the same way as the Integral in 

 (18) for X. For values of a near unity, we find an expansion in powers of q/p of the form 



/5j(a) = 1 + cl 2 tan~V(a 



* 3QP-V(a- 1) jl*-,Ha- I) ♦ ^(a- 1) j.* ....] 



(37) 



while from o near a„ 



P^ia) = /;^(a) * L^ Aa^ - a) (38) 



The graph of/?- l as a function of a is shown In Figure 2, for the case when P/O = 10 , 

 and corresponding tc any initial velocity. When the initial velocity Is zero the curve shrlnlis 

 up to the straight line/3= 1, corresponding to the obvious fact that the centre of the bubble will 

 remain in any displaced position unless it is given an initial velocity. 



Nature of the solution for large n . 



An interesting question, but one difficult to answer satisfactorily without mechanical aid 

 such as the differential analyser, is that of finding numerical solutions of ft 9) for large n. 

 What In fact one would like to Know is whether Initial small Irregularities on the surface become 

 bigger or die out as the rotion proceeds. There seems no doubt that any irregularity limited 

 initially to n small solid angle is unstable, and that the instability increases rapidly as the 

 Initial solid angle decreases. "erhaps the prickly appearance of the bubble in Edgerton's 

 photographs is a manifestation of Instaoility of the type now under discussion; at least there seems 

 no more probable alternative. It may be noted, however, that thi; photographs do not show any pits, 

 out only needles, whereas the theory in its present fjrm indicates both equally. There is clearly 

 a close formal similarity between the Instaoility of very snnall irregularities on the surface of 

 the bubble and ripples en the free surface of water, as the pressure wave strikes. * theory of 

 the magnification of these ripples has been developed by G.l. Taylor, 



The function F(x) as defined by (30) has as its dominant terms, when n is large, 



F(x) = n(ua-*- 3a-^) 

 so that (29) becomes 



^ = ny(4a-'- 3a'^). 



dx'' The 



